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Thought for a couple of seconds

Presenting solution steps clearly and concisely

Great! The area expression is simplified, and it’s independent of 'a'. So, the cross-sectional area is 7/4, and the volume is 21/8 — as we expected. I’ll to present these steps concisely, showing enough work to clarify the process but not overloading with algebra. The final answer should be in Russian too. Let's take it step-by-step:

1. Start by analyzing the geometry and defining the coordinate system for the base.

Simplifying reasoning about cross-section area

To simplify, I’ll assume the diagonals are perpendicular (sin φ = 1), so AC = 2S/BD = 3. The volumes don’t depend on base shape, but I need to explain why. The chosen shape and coordinates like BD & AC don’t affect the final result, so I’ll focus on the simpler approach without overcomplicating things for this level.

Simplifying to a suitable solution approach

The user likely just needs a correct solution. I'll set up a coordinate system with base points: BD along the x-axis, and B and D at the appropriate coordinates. I’ll assume the second diagonal AC is perpendicular to BD for simplicity, making the parallelogram a rhombus. The volume depends only on area and relative positions, so this setup will work for any pyramid with the same base shape.

Simplifying the explanation

I could say: "Let ABCD be a rhombus with perpendicular diagonals. This is possible since only the area of the base, the length of BD, and the height's position through the intersection of the diagonals are given. All such parallelograms can be translated or shifted along the AC diagonal, preserving BD, area, and height, so the volumes remain the same."

Finalizing the solution approach

I'll explain the solution in a more straightforward way: "For convenience, let's choose a coordinate system where diagonals are perpendicular. This doesn't affect the volume, so it's fine. Many Olympiad solutions do this. I'll calculate the height using the plane's equation, then compute the cross-section area. This approach works well for typical student-level problems, and the answer will still be correct."

Volume calculation for pyramid and sectional area

I derived the equations for the pyramid's geometry, including the volume and specific section areas. I computed the height, volume, and area by breaking the section into triangles. The process was a bit heavy on calculations, but it's manageable with the right approach, though I might simplify further.

Considering cross-section area using projection

I’m thinking the area of the cross-section can be calculated by working with projections. When projecting onto the Oyz plane, the lengths and areas are preserved because the projection is parallel to BD. However, I’m unsure if I'll need to factor in any angle adjustments. We’ve already calculated part of the area, but modeling the triangular faces might remove some complexity.

Exploring volume using barycentric coordinates

I’m thinking that we could break the pyramid into smaller tetrahedra (SABO, SBCO, etc.) based on where the plane cuts the edges. Using barycentric coordinates, we can relate the plane's equation to the volumes of these pyramids. We know the final area is 7/4, and symmetry might help simplify calculations.

Calculating the area of quadrilateral AKNM

The quadrilateral AKNM is symmetric across the plane x=0. By reflecting the two triangles AKN and AMM, we find the area of AKNM is twice the area of triangle AKM. We already found that area(AKM) = 7/8, so the quadrilateral's area is 7/4.

Final volume calculation

After computing the cross product of vectors and simplifying, I found the area of triangle AKM to be 7/8. Doubling this gives the quadrilateral's area,

Thought for 12m 43s